Curl-Div Mixed Finite Element Solver
Automated FreeFEM++ Numerical Experiment System — Batch Solver Generation and Execution Framework

English · [中文]

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📋 Requirements

Required Software

• FreeFEM++ 4.0+ — Finite element solver
• Python 3.8+ — Workflow management

Python Dependencies

pip install -r requirements.txt

requirements.txt:

jinja2>=3.1.0    # Template engine
sympy>=1.12      # Symbolic computation

Optional Software

• Ghostscript — EPS → PNG/PDF image format conversion

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🚀 Quick Start

python pipeline.py    # Generate → Solve → Convert

💡 Image conversion requires Ghostscript. If not installed, run:

python pipeline.py --step solve    # Generate → Solve

Execution Example:

Batch generating 24 FreeFEM++ scripts
Using 4 parallel tasks
[1/24] [OK] Dirichlet_Trig_Square/BDM1_P2
...
[24/24] [OK] Magnetic_Trig_Lshaped/BDM2Ortho_P3

Generated: 24 succeeded, 0 failed
Summary: 24/24 succeeded

[SUCCESS] Pipeline completed in 20.3s

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📍 Project Overview

FreeFEM++ Core Functionality

FreeFEM++ excels in 2D mixed finite element solving with mature mesh generation capabilities and rich function space support, efficiently handling complex PDE systems. It handles finite element discretization, linear system solving, and numerical result output.

Other Aspects of Numerical Experiments

Large-scale numerical experiments require more than just the solver:

1. Symbolic computation — Derive partial derivatives, divergence, curl, and source terms from exact solutions for error verification
2. Batch code generation — Automatically generate solver scripts for parameter combinations (boundary conditions × domains × finite element spaces)
3. Image format conversion — Convert FreeFEM++ EPS output to PNG format

Purpose of This Project

This project provides pre-processing and post-processing for FreeFEM++:

Workflow:

1. SymPy symbolic derivation — Given exact solutions, automatically derive derivatives, divergence, curl, and source terms
2. Jinja2 template engine — Inject configuration parameters into FreeFEM++ script templates for batch solver generation
3. Python workflow management — Coordinate generation, solving, and result storage with multi-core parallelism
4. Ghostscript image conversion — Batch convert EPS images to PNG format

Experiment Extension: Adding new experiments only requires defining test function configurations in the function library; the system automatically handles script generation, solving, and result analysis

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📁 Project Structure

rot_div_refactor/
├── pipeline.py                       # Main entry, coordinates three execution stages
├── requirements.txt                  # Python dependency list
├── scripts/                          # Python modules
│   ├── batch_generate.py             # Batch solver script generation
│   ├── run_freefem.py                # FreeFEM++ parallel solving
│   ├── convert_plots.py              # Ghostscript image conversion
│   ├── template_generator.py         # Jinja2 template rendering engine
│   ├── symbolic_derivatives.py       # SymPy symbolic computation
│   ├── function_library.py           # Test function configuration library
│   └── parallel_runner.py            # Parallel framework
├── templates/                        # Jinja2 templates
│   ├── solver.edp.j2                 # FreeFEM++ main solver template
│   └── includes/                     # Sub-templates (mesh, error, output, plot)
│       ├── arrays.idp.j2
│       ├── mesh.idp.j2
│       ├── errors.idp.j2
│       ├── output.idp.j2
│       └── plot.idp.j2
└── output/                           # Output directory

Core File Descriptions:

• function_library.py — Test function configuration library (add new test functions)
• batch_generate.py — FreeFEM++ script generator
• pipeline.py — Main entry and workflow control
• templates/solver.edp.j2 — FreeFEM++ solver script main template
• templates/includes/*.idp.j2 — Sub-templates (mesh generation, error calculation, result output, plotting)

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💻 Usage

Execute Complete Workflow

python pipeline.py                  # Generate + Solve + Convert
python pipeline.py --step generate  # Generate scripts only
python pipeline.py --step solve     # Generate + Solve (no image conversion)

Filter Configurations

python pipeline.py --filter Dirichlet_Trig_Square    # Combination filter
python pipeline.py --filter BDM2                     # All BDM2 spaces

Output Control

python pipeline.py --output workspace    # Specify output directory
python pipeline.py --dpi 300             # High-resolution images

Output Structure

output/
└── {BoundaryCondition}_{Function}_{Domain}/
    └── {FESpace}_{LagrangeSpace}/
        ├── solver.edp           # FreeFEM++ solver
        ├── results.dat          # Numerical solution data
        ├── summary.txt          # Convergence rate report
        └── eps/
            ├── *.eps            # Original images
            └── png/
                └── *.png        # Converted images

Output Example

Convergence Rate Analysis Report (summary.txt):

========================================
Convergence Analysis Report
========================================
Problem: Dirichlet_Trigonometric_Square
Finite Element: BDM1 + P2
Mesh refinements: 4
========================================

========================================
H1 Error of u
========================================
Mesh    Error          Rate
----------------------------------------
  0     1.234e-01   -
  1     3.089e-02   2.00
  2     7.721e-03   2.00
  3     1.930e-03   2.00

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🔧 Extensions

Add Test Functions

Edit scripts/function_library.py:

FUNCTION_LIBRARY = {
    'Dirichlet': {
        'MyFunction': {
            'u1': 'x**2 * (1-x)**2 * y',
            'u2': '-x * y**2 * (1-y)**2',
            'domain': ['Square']
        }
    }
}

Then run:

python pipeline.py --filter MyFunction

Add Finite Element Spaces

Edit FESPACE_COMBINATIONS in scripts/batch_generate.py:

FESPACE_COMBINATIONS = {
    'Dirichlet': [
        ('BDM1', 'P2'),
        ('BDM2', 'P3'),
        ('RT2', 'P2'),    # Add Raviart-Thomas space
    ],
}

Custom Mesh Refinement

Modify the generated solver.edp file or pass FreeFEM++ parameters:

FreeFem++ solver.edp -nref 5    # 5 adaptive refinements

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